LECTURE 2 Actions in Dimension 1 or 2 Suppose Γ is a lattice in a connected, noncompact, simple Lie group G. When M has small dimension, one can sometimes prove that every action of Γ on M is nearly trivial, in the following sense: (2.1) Definition. An action of a group Γ is finite if the kernel of the action is a finite-index subgroup of Γ. 2A. Finite actions (2.2) Remark. 1) It is easy to see that an action of Γ is finite if and only if it factors through an action of a finite group. More explicitly, every finite action of Γ on a space M can be obtained from the following construction: let F be a finite group that acts on M, so we have a homomorphism ϕ1 : F Homeo(M), and let ϕ2 : Γ F be any homomorphism. Then the composition ϕ = ϕ1◦ϕ2 : Γ Homeo(M) defines a finite action of Γ on M. 2) It is known that an action of a finitely generated group on a connected manifold is finite if and only if every orbit of the action is finite. (One direction of this statement is obvious, but the other is not.) It is useful to know that, in certain situations, the existence of a single finite orbit implies that every orbit is finite: (2.3) Proposition. Suppose that every n-dimensional linear representation of every finite-index subgroup of the lattice Γ is finite. If a smooth action of Γ on a connected n-manifold M has a finite orbit, then the action is finite. The proof utilizes the following fundamental result: (2.4) Lemma (Reeb-Thurston Stability Theorem). Suppose the group Λ is finitely generated and acts by C1 diffeomorphisms on a connected manifold M , with a fixed point p. If Λ acts trivially on the tangent space Tp(M), and there is no nontrivial homomorphism from Λ to R, then the action is trivial (i.e., every point of M is fixed by every element of Λ). 9 http://dx.doi.org/10.1090/cbms/109/02
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